code: def _jax_is_indep_analytic(func, *args, **kwargs):
      """Test analytically whether a function is independent of its arguments
      using JAX.
  
      Args:
          func (callable): Function to test for independence
          args (tuple): Arguments for the function with respect to which
              to test for independence
          kwargs (dict): Keyword arguments for the function at which
              (but not with respect to which) to test for independence
  
      Returns:
          bool: Whether the function seems to not depend on it ``args``
          analytically. That is, an output of ``True`` means that the
          ``args`` do *not* feed into the output.
  
      In JAX, we test this by constructing the VJP of the passed function
      and inspecting its signature.
      The first argument of the output of ``jax.vjp`` is a ``Partial``.
      If *any* processing happens to any input, the arguments of that
      ``Partial`` are unequal to ``((),)`.
      Functions that depend on the input in a trivial manner, i.e., without
      processing it, will go undetected by this. Therefore we also
      test the arguments of the *function* of the above ``Partial``.
      The first of these arguments is a list of tuples and if the
      first entry of the first tuple is not ``None``, the input arguments
      are detected to actually feed into the output.
  
      .. warning::
  
          This is an experimental function and unknown edge
          cases may exist to this two-stage test.
      """
      import jax
  
      mapped_func = lambda *_args: func(*_args, **kwargs)
      _vjp = jax.vjp(mapped_func, *args)[1]
      if _vjp.args[0].args != ((),):
          return False
      if _vjp.args[0].func.args[0][0][0] is not None:
          return False
  
      return True
  
file_path: pennylane/pennylane/math/is_independent.py
imports: import warnings
  import numpy
  from autograd.tracer import isbox, new_box, trace_stack
  from autograd.core import VJPNode
  from pennylane import numpy
label_desc: Use image node_id 2 for calling a global function with example usage: _jax_is_indep_analytic(func) and returns: True, False, False